[tnecniv]

The coupling between variables is given by the (quadratic) potential of the system.

I think there is some confusion because they have two tiers of “particles” going on. The first is the masses coupled in the spring-mass system. In that system, each component of x is a particle. However, any specific vector x, in the parlance of thermodynamics, is a single microstate of the system. You can then form a macrostate, I.e., a distribution, of microstate particles, by considering an infinite (or near infinite) number of particles. The dynamics of the macrostate are given by the Fokker-Planck equation, where interactions where both interactions you mention come from the diffusion term only present due to connection with a heat bath.

So the n coupled masses are viewed as a single particle in an abstract system with (stochastic) gradient dynamics.